Zuhair>> This theory can interpret second order arithmetic. And I like to think> of it as a base theory on top of which any stronger set theory can> have its axioms added to it relativized to sets and with set> membership defined as above, so for example one can add all ZFC axioms> in this manner, and the result would be a theory that defines a model> of ZFC, and thus proves the consistency of ZFC. Anyhow this would only> be a representation of those theories in terms of different> primitives, and it is justified if one think of those primitives as a> more natural than membership, or if one think that it is useful to> explicate the later. Moreover this method makes one see the Whole> Ontology involved with set\class theories, thus the bigger picture> revealed! This is not usually seen with set theories or even class> theories as usually presented, here one can see the interplay between> sets and classes (collections of atoms), and also one can easily add> Ur-elements to this theory and still be able to discriminate it from> the empty set at the same time, a simple approach is to stipulate the> existence of atoms that do not represent any object. It is also very> easy to explicate non well founded scenarios here in almost flawless> manner. Even gross violation of Extensionality can be easily> contemplated here. So most of different contexts involved with various> maneuvering with set\class theories can be easily> paralleled here and understood in almost naive manner.>> In simple words the above approach speaks about sets as being atomic> representatives of collections (or absence) of atoms, the advantage is> clearly of obtaining a hierarchy of objects. Of course an atom here> refers to indivisible objects with respect to relation P here, and> this is just a descriptive atom-hood that depends on discourse of this> theory, it doesn't mean true atoms that physically have no parts, it> only means that in the discourse of this theory there> is no description of proper parts of them, so for example one can add> new primitive to this theory like for example the primitive "physical"> and stipulate that any physical object is an atom, so a city for> example would be an atom, it means it is descriptively an atom as far> as the discourse of this theory is concerned, so atom-hood is a> descriptive modality here. From this one can understand that a set is> a way to look at a collection of atoms from atomic perspective, so the> set is the atomic representative of that collection, i.e. it is what> one perceives when handling a collection of atoms as one descriptive> \discursive whole, this one descriptive\discursive whole is actually> the atom that uniquely represents that collection of atoms, and the> current methodology is meant to capture this concept.>> Now from all of that it is clear that Set and Set membership are not> pure mathematical concepts, they are actually reflecting a> hierarchical interplay of the singular and the plural, which is at a> more basic level than mathematics, it is down at the level of Logic> actually, so it can be viewed as a powerful form of logic, even the> added axioms to the base theory above like those of ZFC are really> more general than being mathematical and even when mathematical> concepts are interpreted in it still the interpretation is not> completely faithful to those concepts. However this powerful logical> background does provide the necessary Ontology required for> mathematical objects to be secured and for> their rules to be checked for consistency.>> But what constitutes mathematics? Which concepts if interpreted in the> above powerful kind of logic would be considered as mathematical? This> proves to be a very difficult question. I'm tending to think that> mathematics is nothing but "Discourse about abstract structure", where> abstract structure is a kind of free standing structural universal.> Anyhow I'm not sure of the later. I don't think anybody really> succeeded with carrying along such concepts.>> Zuhair